Π-trivial map

1 Introduction

This page is based on [Ranicki2002]. A map $f:N^n\to M^m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$$f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover with covering map $p:\widetilde{M} \to M$$p:\widetilde{M} \to M$. If $f$$f$ factors through $\widetilde{M}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Bx22oy$$\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$$f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$$f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$$\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a choice of lift $\widetilde{f}$$\widetilde{f}$ is required in order to represent a homology class.

Let $b_1$$b_1$ be a basepoint of $N$$N$. By covering space theory (c.f. [Hatcher2002, Proposition 1.33]) a map $f:N\to M$$f:N\to M$ can be lifted to $\widetilde{M}$$\widetilde{M}$ if and only if $f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$$f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$$q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$ is trivial for $q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$$q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$ the quotient map.

The group $\pi$$\pi$ is well-defined for any choice of lift $\widetilde{f(b_1)}$$\widetilde{f(b_1)}$ since $p:\widetilde{M}\to M$$p:\widetilde{M}\to M$ is a regular covering and changing the basepoint in $\widetilde{M}$$\widetilde{M}$ to a different lift corresponds to conjugating $\pi_1(\widetilde{M},\widetilde{f(b_1)})$$\pi_1(\widetilde{M},\widetilde{f(b_1)})$ by some $g\in \pi_1(M,f(b_1))$$g\in \pi_1(M,f(b_1))$.

2 Definition

Let $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qAQ3yy$$M$ be an $m$$m$-dimensional manifold and let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover. A $\pi$$\pi$-trivial map $f:N^n\to M^m$$f:N^n\to M^m$ is a map from an oriented manifold $N$$N$ with basepoint $b_1$$b_1$ such that the composite

$\displaystyle \xymatrix{ \pi_1(N,b_1) \ar[r]^-{f_*} & \pi_1(M,f(b_1)) \ar[r] & \pi }$

is trivial, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$$\widetilde{f}:N \to \widetilde{M}$.

3 Properties

A map $f:N\to M$$f:N\to M$ that factors through $\widetilde{M}$$\widetilde{M}$ must map all of $N$$N$ to the same sheet of $\widetilde{M}$$\widetilde{M}$, hence the pullback satisfies
$\displaystyle f^*\widetilde{M} \cong N\times \pi.$
Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$$\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$$N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$$\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$.

4 Lifts and paths - two alternative perspectives

Rather than taking a lift as part of the data for a $\pi$$\pi$-trivial map we could instead take an equivalence class of paths in $M$$M$ as is explained in this section. Since $\pi$$\pi$ is the group of deck transformations of $\widetilde{M}$$\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$$\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$$\pi$ with the group structure determined by the action of $\pi$$\pi$ once a choice of lift $\widetilde{f}_{\!\id}$$\widetilde{f}_{\!\id}$ has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a $\pi$$\pi$-trivial map can be thought of as a choice of isomorphism

$\displaystyle \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g \; \; \mathrm{s.t.}\; \widetilde{f}(b_1) = g\widetilde{f}_{\!\id}(b_1).\end{array}$
Let $b$$b$ be a basepoint of $M$$M$. The set of homotopy classes of paths from $b$$b$ to $f(b_1)$$f(b_1)$ is non-canonically isomophic to $\pi_1(M,b)$$\pi_1(M,b)$. An isomorphism is defined by a choice of path $[w_{id}]$$[w_{id}]$ to represent the identity element:
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where $*$$*$ denotes concatenation of paths and $w_{id}^{-1}$$w_{id}^{-1}$ is the path $w_{id}$$w_{id}$ in reverse. Let $\widetilde{b}$$\widetilde{b}$ be a basepoint of $\widetilde{M}$$\widetilde{M}$ that is a lift of $b$$b$. Define an equivalence relation $\sim$$\sim$ on this set by saying
$\displaystyle [w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).$
The above isomorphism given by choosing $[w_{id}]$$[w_{id}]$ descends to give an isomorphism
$\displaystyle \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,$
where we use the same choice of path $[w_{id}]$$[w_{id}]$ to identify $\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})$$\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})$ with $\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$$\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$. Thus a choice of lift $\widetilde{f}:N\to \widetilde{M}$$\widetilde{f}:N\to \widetilde{M}$ corresponds to a choice of homotopy class of paths from $b$$b$ to $f(b_1)$$f(b_1)$ modulo $\pi_1(\widetilde{M})$$\pi_1(\widetilde{M})$. A choice of lift $\widetilde{b}$$\widetilde{b}$ defines a bijection of sets
$\displaystyle \{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$
as follows. Given a choice of lift $\widetilde{f}$$\widetilde{f}$ choose any path $\widetilde{w}:I \to \widetilde{M}$$\widetilde{w}:I \to \widetilde{M}$ from $\widetilde{b}$$\widetilde{b}$ to $\widetilde{f}(b_1)$$\widetilde{f}(b_1)$. Take the equivalence class of $p(\widetilde{w})$$p(\widetilde{w})$ which is a path in $M$$M$ from $b$$b$ to $f(b_1)$$f(b_1)$. Conversely given a choice of class
$\displaystyle [w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$
choose any representative $w:I \to M$$w:I \to M$. This lifts uniquely to a path $\widetilde{w}$$\widetilde{w}$ starting at $\widetilde{b}$$\widetilde{b}$. Define a lift $\widetilde{f}$$\widetilde{f}$ by setting $\widetilde{f}(b_1):= \widetilde{w}(1)$$\widetilde{f}(b_1):= \widetilde{w}(1)$. Note this map is well-defined since different choices of representative $w$$w$ may differ by elements of $\pi_1(\widetilde{M},\widetilde{b})$$\pi_1(\widetilde{M},\widetilde{b})$ but their lifts will still end at the same point.

To sum up we have the following diagram of non-canonical isomorphisms and bijections

$\displaystyle \xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \\ & \pi & }.$
Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups.

Since an oriented cover comes with a choice of lift $\widetilde{b}$$\widetilde{b}$ as part of the data a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a $\pi$$\pi$-trivial map.

5 Examples

Let $f:S^n\looparrowright M^{2n}$$f:S^n\looparrowright M^{2n}$ be an immersion and let $\widetilde{M}$$\widetilde{M}$ be the universal cover of $M$$M$, let $s\in S^n$$s\in S^n$ and $b\in M$$b\in M$ be basepoints. For $n>1$$n>1$, $\pi_1(S^n)=0$$\pi_1(S^n)=0$ so $f$$f$ lifts to $\widetilde{M}$$\widetilde{M}$. An immersion $f:S^n\looparrowright M^{2n}$$f:S^n\looparrowright M^{2n}$ is a $\pi_1(M)$$\pi_1(M)$-trivial immersion as soon as a lift $\widetilde{f}:S^n \looparrowright \widetilde{M}$$\widetilde{f}:S^n \looparrowright \widetilde{M}$ has been prescribed or, alternatively, once a homotopy class of paths $w:I\to M$$w:I\to M$ from $b$$b$ to $f(s)$$f(s)$ has been prescribed. A pair $(f:S^n\looparrowright M^{2n}, w: I\to M | w(0)=b \; \text{and} \; w(1)=f(s))$$(f:S^n\looparrowright M^{2n}, w: I\to M | w(0)=b \; \text{and} \; w(1)=f(s))$ is often called a pointed immersion in the literature (See, for example, [Lück2001, Section 4.1]).